When \(p=1\) , the circle decomposes into a union of geodesics lines (arcs), known as “transport rays”, which intersect (potentially) only at their endpoints. On each such transport ray, the optimal transport map is monotone; see details in [1]} and [2]}. This alone precludes the crossing scenario completely. The overlapping scenario is also not possible: it would require all intervals \(I_m, I_j, I_n\) , and \(I_l\)  to be on the same transport ray, on which overlapping would then contradict monotonicity.
